Question

P(x) = -0.3x² + 75x - 2000, where x represents the selling price.

1. At what price should the stands be sold to earn the maximum profit?

2. According to the function given, what is the maximum profit that the company can make?

3. What are the break-even points (the selling prices for which the profit is 0)? Give the answer to the nearest cent.​

Answer 1

Below in bold.

Explanation:

P(x) = -0.3x² + 75x - 2000

Convert this to vertex form:

p(x) = -0.3(x^2 - 250)^2 - 2000

= -0.3[(x - 125)^2 - 125^2] - 2000

= -0.3[(x - 125)^2 - 15625] - 2000

= -0.3(x - 125)^2 + 4687.5 - 2000

= -0.3(x - 125)^2 + 2687.5

2687.5 is the maximum value of the profit.

1. This corresponds to a selling price of $125 for the stands.

2. Maximum profit is $2687.5.

3. At the breaking points P(x) = 0:

-0.3(x - 125)^2 + 2687.5 = 0

(x - 125)^2 = 2687.5 / -0.3 = 8958.33

x - 125 = +/- sqrt( 8958.33)

x = 125 +/- sqrt( 8958.33)

x = 30.352, 219.648,

So, the breaking even points are $30.35 and $219.65 to nearest cent.